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Introduction to Hodge Theory

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Introduction to Hodge Theory INTRODUCTION TO HODGE THEORY DANIEL MATEI SNSB 2008 Abstract. This course will present the basics of Hodge theory aiming to familiarize students with an important technique in complex and algebraic geometry. We start by reviewing complex manifolds, Kahler manifolds and the de Rham theorems. We then introduce Laplacians and establish the connection between harmonic forms and cohomology. The main theorems are then detailed: the Hodge decomposition and the Lefschetz decomposition. The Hodge index theorem, Hodge structures and polariza- tions are discussed. The non-compact case is also considered. Finally, time permitted, rudiments of the theory of variations of Hodge structures are given. Date: February 20, 2008. Key words and phrases. Riemann manifold, complex manifold, deRham cohomology, harmonic form, Kahler manifold, Hodge decomposition. 1 2 DANIEL MATEI SNSB 2008 1. Introduction The goal of these lectures is to explain the existence of special structures on the coho- mology of Kahler manifolds, namely, the Hodge decomposition and the Lefschetz decom- position, and to discuss their basic properties and consequences. A Kahler manifold is a complex manifold equipped with a Hermitian metric whose imaginary part, which is a 2-form of type (1,1) relative to the complex structure, is closed. This 2-form is called the Kahler form of the Kahler metric. Smooth projective complex manifolds are special cases of compact Kahler manifolds. As complex projective space (equipped, for example, with the Fubini-Study metric) is a Kahler manifold, the complex submanifolds of projective space equipped with the induced metric are also Kahler. We can indicate precisely which members of the set of Kahler manifolds are complex projective, thanks to Kodaira’s theorem: Theorem 1.1. A compact complex manifold admits a holomorphic embedding into com- plex projective space if and only if it admits a Kahler metric whose Kahler form is of integral class. We are essentially interested in the class of Kahler manifolds, without particularly emphasising projective manifolds. The reason is that our goal here is to establish the existence of the Hodge decomposition and the Lefschetz decomposition on the cohomology of such a manifold, and for this, there is no need to assume that the Kahler class is integral. However, the Lefschetz decomposition will be defined on the rational cohomology only in the projective case, and this is already an important reason to restrict ourselves, later, to the case of projective manifolds. If X is a complex manifold, the tangent space to X at each point x is equipped with a complex structure Jx. The data consisting of this complex structure at each point is what is known as the underlying almost complex structure. The Jx provide a decomposition 1,0 0,1 (1.1) TxX ⊗ C = TxX ⊕ TxX , 1,0 where TxX is the vector space of complexified tangent vectors u such that Jxu = iu 0,1 1,0 and TxX is the complex conjugate of TxX . From the point of view of the complex structure, i.e. of the local data of holomorphic coordinates, the vector fields of type (0, 1) are those which kill the holomorphic functions. The decomposition (1.1) induces a similar decomposition on the bundles of complex differential forms (1.2) ΛkT ∗X ⊗ C = Λp,qT ∗X, p+Mq=k where Λp,qT ∗X = ΛpT ∗X1,0 ⊗ ΛqT ∗X0,1 and T ∗X ⊗ C = T ∗X1,0 ⊕ T ∗X0,1, is the dual decomposition of (1.1). INTRODUCTIONTOHODGETHEORY 3 The decomposition (1.2) has the property of Hodge symmetry Λp,qT ∗X = Λq,pT ∗X, where complex conjugation acts naturally on ΛkT ∗X ⊗ C. k If we let E (X)C denote the space of complex differential forms of degree on X, i.e. the C∞-sections of the vector bundle ΛkT ∗X ⊗ C, then we also have the exterior differential k k+1 d : E (X)C →E (X)C, which satisfies d ◦ d = 0. We then define the kth de Rham cohomology group of X by k k Ker(d : E (X)C →E +1(X)C) Hk(X, C)= . Im(d : Ek−1(X)C →Ek(X)C) The main theorem proved in these notes is the following. Theorem 1.2. Let Hp,q(X) ⊂ Hk(X, C) be the set of classes which are representable by a closed form α which is of type (p,q) at every point x in the decomposition (1.2). Then we have a decomposition (1.3) Hk(X, C)= Hp,q(X). p q k +M= Note that by definition, we have the Hodge symmetry Hp,q(X)= Hq,p(X) where complex conjugation acts naturally on Hk(X, C)= Hk(X, R) ⊗ C. Here Hk(X, R) is defined by replacing the complex differential forms by real differential forms in the above definition. This theorem immediately gives constraints on the cohomology of a Kahler manifold, which reveal the existence of compact complex manifolds which are not Kahler. For example, the decomposition (1.3) and the Hodge symmetry imply that the dimensions k bk(X) := dimC H (X, C), called the Betti numbers, are even for odd k, property not satisfied by Hopf surfaces. Example 1.3. The Hopf surfaces are the quotients of C2 \ {0} by the fixed-point-free action of a group isomorphic to Z, where a generator g acts via g(z1, z2) = (λ1z1, λ2z2) where the λi, are non-zero complex numbers of modulus strictly less than 1. These surfaces are compact, equipped with the quotient complex structures, and their π1 is isomorphic to Z since C2 \ {0} is simply connected. Thus, their first Betti number is equal to 1, which implies that they are not Kahler. The Lefschetz decomposition is another decomposition of the cohomology of a compact Kahler manifold X, this time of topological nature. It depends only on the cohomology class of the Kahler form [ω] ∈ H2(X, R). The exterior product on differential forms satisfies Leibniz’ rule d(α ∧ β)= dα ∧ β + (−1)deg αα ∧ dβ, so the exterior product with ω sends closed forms (i.e. forms killed by d) to closed forms and exact forms (i.e. forms in the image of d) to exact forms. Thus it induces an operator, called the Lefschetz operator, L : Hk(X, R) → Hk+2(X, R). 4 DANIEL MATEI SNSB 2008 The following theorem is sometimes called the hard Lefschetz theorem. Theorem 1.4. For every k ≤ n = dim X, the map (1.4) Ln−k : Hk(X, R) → H2n−k(X, R) is an isomorphism. Remark 1.5. Note that the spaces on the right and on the left are of the same dimension by Poincare duality, which is valid for all compact oriented manifolds. A very simple consequence of the above isomorphism is the following result, which is an additional topological constraint satisfied by Kahler manifolds. Corollary 1.6. The morphism L : Hk(X, R) → Hk+2(X, R) is injective for k ≤ n = dim X. Thus, the odd Betti numbers b2k−1(X) increase with k for 2k − 1 ≤ n, and similarly, the even Betti numbers b2k(X) increase for 2k ≤ n. An algebraic consequence of Lefschetz’ theorem is the Lefschetz decomposition, which as we noted earlier is particularly important in the case of projective manifolds. Let us define the primitive cohomology of a compact Kahler manifold X by k n−k+1 k 2n−k+2 H (X, R)prim := Ker(L : H (X, R) → H (X, R)) for k ≤ n. One can extend this definition to the cohomology of degree >n by using the isomorphism (1.4). Theorem 1.7. The natural map k−2r k i : H (X, R)prim → H (X, R) k−M2r≥0 r (αr) → L αr is an isomorphism for k ≤ n. X Once again, we can extend this decomposition to the cohomology of degree > n by using the isomorphism (1.4). Let us now express the main principle of Hodge theory, which has immense applications. The study of the cohomology of Kahler manifolds and the proof of the Theorems (eq:ihdec) and (1.4), which are the main content of these lectures, are among the most important applications, but the principle applies in various other situations. We restrict ourselves here to giving an explanation of the main idea, which is the notion of a harmonic form, and the application of the theory of elliptic operators which makes it possible to represent the cohomology classes by harmonic forms, but we will omit the proof of the fundamental theorem on elliptic operators, which uses estimations and notions from analysis (Sobolev spaces), which are in different directions from the aims of this book. The delicate point consists in passing from spaces of L2 differential forms, in which the Hodge decomposition is algebraically obvious, to spaces of C∞ differential forms. One of the problems we INTRODUCTIONTOHODGETHEORY 5 encounter is the fact that the operators considered here are differential operators, and thus do not define continuous operators on the spaces of L2 forms. The idea that we want to explain here is the following: using the metric on X, we can define the L2 metric on the spaces of differential forms (α, β)L2 = hα, βix Vol, ZX where α, β are differential forms of degree k and the scalar product hα, βix at a point x ∈ X is induced by the evaluation of the forms at the point x and by the metric at the point x. The operator d : Ek(X) →Ek+1(X) is a differential operator, and we can construct its formal adjoint d∗ : Ek(X) →Ek−1(X), which is also a differential operator, and satisfies the identity ∗ (α, dβ)L2 = (d α, β)L2 , for α ∈Ek(X), β ∈Ek−1(X).
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